The Physics of Dance George Gollin UIUC Dede

The Physics of Dance George Gollin, UIUC Dede Cole, Centennial HS Saturday Physics Honors Program University of Illinois at Urbana-Champaign November 10, 2001

Today. . . • Introduction • Some physics vocabulary • Net force: adding forces • More physics vocabulary • Static balance • Balance while turning • The physics of turns • Jumps (and black holes. . . ) 2

Introduction How’d we get ourselves into this? • George’s story • Dede’s story Some statements about dance technique can be restated using a physics vocabulary… Lifts “feel” the way they do for the same reason that a large star will collapse to form a black hole. . . 3

Some Physics • Velocity ( ): Vocabulary how fast and in what direction something is moving. • Momentum( ): The magnitude of an object’s momentum gives an indication of how hard it is to bring to a stop. • Force: Just like you’d think. It’s a push, characterized by a magnitude (how hard) and a direction. 4

Force changes momentum • The harder you push the more rapidly the momentum changes. • The longer you push the greater the total change in momentum. • The equation: A force can change an object’s speed and/or direction of motion. 5

Adding forces Cart feels two forces: • gravity: downwards • push from track: up & to-the-left The net force causes the change in the cart’s momentum. Forces on a dancer: gravity (down), support from floor (up), friction from floor (sideways) 6

More Physics Vocabulary • Angular velocity ( ): how fast something spins and in what direction its rotation axis points. • Angular momentum( ): The magnitude of an object’s angular momentum gives an indication of how hard it is to bring to a stop. 7

Even More Physics Vocabulary • Large rotational inertia ( ), spinning slowly or small rotational inertia, spinning rapidly… both are hard to stop. • Angular velocity • angular momentum • rotational inertia 8

Still More Physics Vocabulary • Centrifugal force: Effective force pushing something to the outside of a turn. 9

Even Still More Physics Vocabulary • Torque ( ): “off center” forces which can cause something to spin. 10

About torque. . . Torque changes angular momentum: • The harder you push, the more quickly the object “spins up. ” • The further from the spin axis you apply the forces, the more quickly the object spins up. • The longer you push, the greater the total change in angular momentum. • If no torque is applied to a rigid object, its spin axis doesn't wobble. 11

Static Balance We all know what it means. . . in physics terms: • Net force is zero (otherwise your momentum would change: you might fall) • Net torque is zero (otherwise your angular momentum would change: you might tip over) 12

Center of gravity Just like you'd think. You can balance around this point. Reason for the name: net effect of gravity on an object (a dancer) is the same as if it were only acting at the c. g. , and not on all parts of the object. 13

Forces on a Balanced Dancer Gravity (pulls down, acts at c. g. ) + support from floor (pushes up, acts at feet) Keith Roberts (ABT) in The Rite of Spring, photo by Round (New York) Forces sum to zero; both forces "point through" the dancer's center of gravity, so torque is zero.

More complicated example Forces on each dancer sum to zero. Though some forces are “off-center” torques also sum to zero. Sandra Brown and Johann Renvall (ABT) in Airs, 15 photo by Nancy Ellison

Staying Balanced Staying balanced: center of gravity remains directly above the area of contact with the floor. (Otherwise : you'd tip over. ) 16

Staying Balanced Smaller floor contact area (en pointe, for example) makes balance more difficult. Maintaining balance: • shift foot to move floor contact area • adjust arms/legs/torso to move c. g. 17

Balance While Turning Two distinct "physics domains" here: • slow turns: maintain static balance • fast turns: rotation axis shouldn't wobble (much) 18

Balance While Turning Not all positions which are statically balanced can be held in a turn. An example: Centrifugal forces exert a torque, throwing the dancer off balance. 19

Balance While Turning Shift center of gravity to maintain dynamic (but not static) balance: 20

The Physics of Turns Reminders: • • • is angular momentum is rotational inertia is angular velocity is torque 21

The Physics of Turns indicates how hard it is to start something spinning (just as mass is an indication of how hard it is to start something moving). : if no torque is applied constant. must be . . . if no torque is applied: • Decreasing will increase • Increasing will decrease 22

Calculating Rotational Inertia • Greater mass greater. (Doubling the mass doubles. ) • Mass placed farther from rotation axis greater. (Doubling the distance quadruples. ) • • (one mass) (several masses) 23

Increasing Decreases Spin Rate quadruples in the figure. Because of “conservation of angular momentum” you can change the speed of a turn when en pointe by extending or retracting arms and legs. No torque is involved! 24

Increasing Decreases Spin Rate 25

Friction and Torque Friction between the floor and your foot (or feet) can also generate a torque. Swinging a leg going into a turn works like this: You push on the floor one way, the floor pushes back the other way. 26

The Physics of Jumps Gravitational force depends on mass: • More massive greater gravitational pull • Mass is to gravity like electric charge is to electrostatics. In electrostatics: the larger the electric charge something has, the greater the force it feels in an electric field. Momentum also depends on mass: the greater your mass, the harder it is to stop moving at a particular speed. (Recall: p = mv) 27

The Physics of Jumps This is very curious. Why should an object's momentum/velocity relationship have anything at all to do with the gravitational pull on that object? (Doubling an object's mass both doubles its momentum and doubles the strength of the gravitational force it experiences. ) 28

The Physics of Jumps The equivalence of "gravitational" and "inertial" mass gives rise to Einstein's General Theory of Relativity. Black holes, curved spacetime, and so on! If they were not equivalent, dance technique associated with jumps would need to be different from what it actually is. 29

Gravity. . . …is the only force acting on a dancer in midair. Recall: net effect of gravity is the same as if it were only acting at the dancer’s center of gravity. Gravitational force: ( is constant) Force and momentum: 30

Gravitational acceleration. . . …is independent of mass. Anything you drop speeds up at the same rate; anything thrown straight up slows down at the same rate. Vertical speed changes by 32 feet per second (about 22 mph) for each second spent in the air. 31

Gravity Only Influences Vertical (Not Horizontal) Motion Velocity of a dancer’s center of gravity vs. time during a jump: Dancer’s vertical motion slows, stops, changes direction as dancer rises, then falls. 32

Trajectory in a Jump Height vs. time is a parabola; horizontal distance vs. time is a straight line: 33

Trajectory in a Jump The dancer’s center of gravity moves along a trajectory which is also a parabola. For a 2 foot rise in the height of the c. g. and a 10 ft/sec horizontal velocity we see the following: 34

Illusions in a Jump By raising his/her legs (changing the distance between the c. g. and his/her head). . . The dancer can create an illusion of floating. . . 35

Illusions in a Jump The dancer can create an illusion of floating. . . The vertical motion of the head is smaller than the vertical motion of the center of gravity. 36

Illusions in a Jump Half the duration of the jump is spent between the points indicated by the arrows; the height of the dancer's center of gravity only changes by 6 inches during this time.

Dance is an Art • Physics does not “explain” dance • The reaction of the audience to dance is not “enhanced” by knowing the physics • There are interesting parallels between some ideas of current interest in the humanities and those of modern physics (e. g. the blurring of boundaries between a thing and its environment) • It is interesting to see where dance/physics vocabularies overlap. 38

With More Time. . . Time permitting, it would be fun to investigate some of the following: • dancing with a partner • effects relating to body size • impacts and stress injuries • the connection between “good technique” and injury prevention 39

Acknowledgements and References Dance faculty at UIUC: Linda Lehovec, Rebecca Nettl-Fiol, Sean Sullivan, Renée Wadleigh, Erica Wilson-Perkins Dance faculty at Hope College (Michigan): Linda Graham More information: Kenneth Laws and Cynthia Harvey, Physics, Dance, and the Pas de Deux, Schirmer Books, New York, 1994. http: //www. uiuc. edu/ph/www/g-gollin/dance_physics. html 40